Different types of multifractal measures in separable metric spaces and their applications

Najmeddine Attia; Bilel Selmi; Najmeddine Attia; Bilel Selmi

doi:10.3934/math.2023650

AIMS Mathematics

2023, Volume 8, Issue 6: 12889-12921. doi: 10.3934/math.2023650

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Research article

Different types of multifractal measures in separable metric spaces and their applications

Najmeddine Attia ^{1
,
,},
Bilel Selmi ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, PO. Box : 400 Al-Ahsa 31982, Saudi Arabia
2.
Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000-Monastir, Tunisia

Received: 18 January 2023 Revised: 25 February 2023 Accepted: 15 March 2023 Published: 31 March 2023
MSC : 28A78, 28A80

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.
- generalized Hausdorff measure,
- generalized packing measure,
- Henstock-Thomson variation,
- regularities,
- densities,
- doubling measures
Citation: Najmeddine Attia, Bilel Selmi. Different types of multifractal measures in separable metric spaces and their applications[J]. AIMS Mathematics, 2023, 8(6): 12889-12921. doi: 10.3934/math.2023650

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Abstract

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.

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